S˘i’lnikov-type orbits of Lorenz-family systems

This paper studies the Lorenz-family system which is known to establish a topological connection among the Lorenz, Chen and Lu¨ systems in the parametric space. The existence of S˘i’lnikov heterclinic orbits is proved using an undetermined coefficient method. As a consequence, the S˘i’lnikov criterion along with some technical conditions guarantees that the Lorenz-family system has both Smale horseshoes and horseshoe type of chaos. It is this heteroclinic orbit that determines the geometric structure of the corresponding chaotic attractor.